Number of generators of ideals in Jordan cells of the family of graded Artinian algebras of height two

Abstract

We let $A=R/I$ be a standard graded Artinian algebra quotient of $R={\sf k}[x,y]$, the polynomial ring in two variables over a field ${\sf k}$ by an ideal $I$, and let $n$ be its vector space dimension. The Jordan type $P_\ell$ of a linear form $\ell\in A_1$ is the partition of $n$ determining the Jordan block decomposition of the multiplication on $A$ by $\ell$ -- which is nilpotent. The first three authors previously determined which partitions of $n=\dim_{\sf k}A$ may occur as the Jordan type for some linear form $\ell$ on a graded complete intersection Artinian quotient $A=R/(f,g)$ of $R$, and they counted the number of such partitions for each complete intersection Hilbert function $T$ arXiv:1810.00716.\par We here consider the family $\mathrm{G}_T$ of graded Artinian quotients $A=R/I$ of $R={\sf k}[x,y]$, having arbitrary Hilbert function $H(A)=T$. The Jordan cell $\mathbb V(E_P)$ corresponding to a partition $P$ having diagonal lengths $T$ is comprised of all ideals $I$ in $R$ whose initial ideal is the monomial ideal $E_P$ determined by $P$. These cells give a decomposition of the variety $\mathrm{G}_T$ into affine spaces. We determine the generic number $\kappa(P)$ of generators for the ideals in each cell $\mathbb V(E_P)$, generalizing a result of arXiv:1810.00716. In particular, we determine those partitions for which $\kappa(P)=\kappa(T)$, the generic number of generators for an ideal defining an algebra $A$ in $\mathrm{G}_T$. We also count the number of partitions $P$ of diagonal lengths $T$ having a given $\kappa(P)$. A main tool is a combinatorial and geometric result allowing us to split $T$ and any partition $P$ of diagonal lengths $T$ into simpler $T_i$ and partitions $P_i$, such that $\mathbb V(E_P)$ is the product of the cells $\mathbb V(E_{P_i})$, and $T_i$ is single-block: $\mathrm{G}_{T_i}$ is a Grassmannian.